Impulse parameter and a new equivalence between 123- and 132-avoiding permutations

Abstract

In this paper, we enumerate permutations [Formula: see text] according to the number of indices [Formula: see text] such that [Formula: see text], where [Formula: see text] and [Formula: see text] is a fixed positive integer. We term such an index [Formula: see text] an [Formula: see text]-impulse since it marks an occurrence where the bargraph representation of [Formula: see text] rises above (or to the same level as) the horizontal line [Formula: see text]. We find an explicit formula for the distribution as well as a formula for the total number of [Formula: see text]-impulses in all permutations of [Formula: see text]. Comparable distributions are also found for the [Formula: see text]-avoiding permutations of [Formula: see text], where [Formula: see text] is a pattern of length three. Two markedly different distributions emerge, one for [Formula: see text] and another for the remaining patterns in [Formula: see text]. In particular, we obtain a new equidistribution result between 123- and 132-avoiding permutations. To prove our results, we make use of multiple arrays and systems of functional equations, employing the kernel method to solve the system in the case [Formula: see text]. We also provide a combinatorial proof of the aforementioned equidistribution result, which actually applies to a more general class of multi-set permutations.